ECE 656 Exam 2: Fall 2013 September 23, 2013 Mark Lundstrom Purdue University (Revised 9/25/13)

Transcription

1 NAME: PUID: : ECE 656 Exam : September 3, 03 Mark Lundstrom Purdue University (Revised 9/5/3) This is a closed book exam. You may use a calculator and the formula sheet at the end of this exam. There are three equally weighted questions. To receive full credit, you must show your work (scratch paper is attached). The exam is designed to be taken in 50 minutes. Be sure to fill in your name and Purdue student ID at the top of the page. DO NOT open the exam until told to do so, and stop working immediately when time is called. The last page is an equation sheet, which you may remove, if you want. 30 points possible, 0 per question ) points per part 0 points total a) 5 points b) 5 points 3a) 4 points 3b) 4 points 3c) points ECE- 656

2 Exam : ECE 656 Answer the five multiple choice questions below by choosing the one, best answer..) What are the important scattering mechanisms for electrons in undoped Si at room temperature? a) ADP and PZ intravalley scattering b) ADP intravalley scattering and intervalley phonon scattering. c) ADP intravalley and POP inter valley scattering. d) ADP intravalley, alloy and neutral defect scattering. e) ADP intravalley and plasmon scattering..) What is the most important scattering mechanism in undoped GaAs at room temperature? a) IV scattering b) ODP intravalley scattering c) ADP intravelley scattering d) POP intravalley scattering e) PZ intravalley scattering.3) How does the energy relaxation time generally compare to the other characteristic times? a) τ E τ m,τ E τ b) τ E τ m,τ E > τ c) τ E > τ m,τ E τ d) τ E > τ m,τ E > τ e) τ E < τ m,τ E τ..4) How does the ADP momentum relaxation time vary with temperature? a) Approximately independent of temperature. b) Increases as temperature increases. c) Decreases as temperature increases. d) Displays a maximum at the Debye temperature. e) Displays a minimum at the Debye temperature..5) How does the II momentum relaxation time vary with temperature? a) Approximately independent of temperature. b) Increases as temperature increases. c) Decreases as temperature increases. d) Displays a maximum at the Debye temperature. e) Displays a minimum at the Debye temperatur ECE- 656

3 Exam : ECE 656 ) The material, In 0.53 Ga 0.47 As is an important semiconductor because it is lattice matched to InP. It is a direct bandgap material with a bandgap of E GΓ = 0.75 ev and an effective mass of m n * m 0 = It contains heavy mass, upper valleys located at an energy of ΔE Γ L = 0.55 ev above the Γ valley minimum. It has an optical phonon energy of 3 mev. The following two questions concern electron scattering in undoped In 0.53 Ga 0.47 As at room temperature. a) Sketch the scattering rate vs. energy for electrons in the Γ valley from E = 0 (bottom of the Γ valley) to E = 0.9 ev. Your sketch should label each of the main phonon scattering processes, the critical energies, and the relative magnitudes of the processes. (DO NOT add all of the processes to get the total scattering rate, just sketch each one separately.) SHOW YOUR WORK. ECE

4 Exam : ECE 656 b) Sketch the intervalley scattering rate vs. energy for electrons in the L valley from E = 0 (bottom of the Γ valley) to E = 0.9 ev. Your sketch should label each of the main phonon scattering processes, the critical energies, and the relative magnitudes of the processes. (DO NOT add all of the processes to get the total scattering rate, just sketch each one separately.) SHOW YOUR WORK. Note: There is also intravalley scattering within the L- valley, but you are not being asked to plot this. ECE

5 Exam : ECE 656 3) This problem involves ODP intravalley scattering of electrons in graphene. Recall that the dispersion of graphene is E( k) = ±υ F k x + k y = ±υ F k, and the density- of- states is D( E) = E E > 0. π υ F (Note that the factor of in the above expression comes from the valley degeneracy of for graphene.) The transition rate is given by FGR as: S p, ( p ) = π H ( ), p, p δ E E ω 0 where p refers to an electron in the plane of the graphene sheet (the x- y plane). The ODP scattering potential is U S = D 0 u β The lattice vibration is written as u β ρ ( ) = Aβ e ±i β ρ, where ρ is a vector in the x- y plane and β is a phonon wavevector in the x- y plane. Following the procedure in the text, we write the amplitude of the phonon wavevector as A β = N 0 + ρ m A ω 0, where ρ m is the mass density in Kg/m. Answer the following three questions. Draw a box around your answers. For this problem, you should just assume simple plane waves; do not worry about the two- component wavefunction. ECE

6 Exam : ECE 656 3a) Derive an expression for the transition rate, S p (, p ) due to intravalley phonon emission. Assume that electrons are in the graphene conduction band, with E > > 0. You must show your work. 3b) Using your result from part 3a) derive an expression for the intravalley scattering rate due to intravalley phonon emission. HINT: You do not need to explicitly consider momentum conservation for this problem. You must show your work. continued on next page ECE

7 Exam : ECE 656 3c) Using your results from above, derive the energy relaxation time assuming energetic carriers for which phonon emission dominates. You must show your work. ECE

8 SCRATCH PAPER ECE

9 SCRATCH PAPER ECE

10 ECE#656&Key&Equations&(Weeks&3#5)& % Physical&constants:%% = [ J-s] %% m 0 = [ kg] % k B = [ J/K] % q = [ C] % ε 0 = [ F/cm] % Density&of&states&in&k#space:& D:% N k = L π ( ) = L π % D:% N k = ( A 4π ) = A π % 3D:% N k = ( Ω 8π ) = Ω 4π 3 % % Density&of&states&in&energy&(parabolic&bands,&per&length,&area,&or&volume):& D D ( E) = g v π m * ( E ε ) % D E D ( ) = g V m * π % D 3D E m ( ) * m * ( E E C ) = g v π 3 % % Fermi&function&and&Fermi#Dirac&Integrals:& f 0 ( E) = ( + e E E F ) & k B T η F j ( η F ) j dη df = Γ( j + ) % F + e η η F j ( η F ) e η j η F << 0 % = F dη j & 0 F Γ(n) = ( n )!%(n%an%integer)%% Γ( / ) = π %% % Γ( p + ) = pγ( p) % % Scattering:& S p, p ( ) = π H τ p p, p % % ( ) == S ( p, p ) p, δ ( E E ΔE ) % τ m p + H p, = p Ω e i p r U S ( r )e i p r dr % ( p ) Δp z % % p p ( ) ΔE % ( ) = S p, p, p z0 τ E ( ) = S p, p, ADP:% K β = β D A % ODP:% K β = D0 % PZ:% K β = ( qepz κ S ε 0 ) ρq ω % POP:% K β = 0 S p, p δ ( ) = π Ωρ ω K β ( ) p, p± δ E E ω β β τ = + = π τ abs τ ems τ = = π τ m N ω + δ p, p± δ E E ω β D A k B T c l ( ) % υβ δ β ±cosθ + p ω β υβ % D 3D ( E) %(ADP)% L D = κ S ε 0 k B T L q n 0 % ( ) D O ρω 0 N + 0 D 3D E ± ω 0 % N 0 = e ω o k B T L %%(ODP)% E 0 κ 0 β κ 0 ε 0 κ % Mark%Lundstrom% % 8/9/03% % ECE